All Questions
5
questions
5
votes
2
answers
364
views
How to find range $a_{75}$ of the term of the series $a_n=a_{n-1}+ {1 \over {a_{n-1}}} $ [duplicate]
If $a_1=1$ and for n>1$$a_n=a_{n-1}+ {1 \over {a_{n-1}}} $$
$a_{75}$ lies between
(a) (12,15)
(b) (11,12)
(c) (15,18)
Now , in this question, I rewrote, $a_n-a_{n-1} = {1 \over {a_{n-1}}}$, to ...
- 1,766
12
votes
2
answers
937
views
$\sum\limits_{i=1}^n \frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod \limits_{j=1}^nx_j}} \ge 1$, for all $x_i>0.$
Can you prove the following new inequality? I found it experimentally.
Prove that, for all $x_1,x_2,\ldots,x_n>0$, it holds that
$$\sum_{i=1}^n\frac{x_i}{\sqrt[n]{x_i^n+(n^n-1)\prod\limits _{j=...
- 2,040
5
votes
3
answers
450
views
Non-induction proof of $2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1$
Prove that $$2\sqrt{n+1}-2<\sum_{k=1}^{n}{\frac{1}{\sqrt{k}}}<2\sqrt{n}-1.$$
After playing around with the sum, I couldn't get anywhere so I proved inequalities by induction. I'm however ...
- 2,916
6
votes
3
answers
208
views
Algebraic inequality $\sum \frac{x^3}{(x+y)(x+z)(x+t)}\geq \frac{1}{2}$
The inequality is
$$\frac{x^3}{(x+y)(x+z)(x+t)}+\frac{y^3}{(y+x)(y+z)(y+t)}+\frac{z^3}{(z+x)(z+y)(z+t)}+\frac{t^3}{(t+x)(t+y)(t+z)}\geq \frac{1}{2},$$
for $x,y,z,t>0$.
It originates from a 3-D ...
- 1,250
3
votes
2
answers
144
views
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
As the title says I need to prove the following by induction:
$$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$
When trying to prove that P(n+1) is true if P(n) is, then I ...
- 706